Envelope representation of Hamilton–Jacobi equations from spin glasses

Mean-field theories of spin glasses

The theory of the random Ising model formulated in terms of the distribution function of the effective field in the pair and in the cluster approximations is reviewed. An integral equation for the distribution function is derived. The integral equation has a solution for the paramagnetic state, a solution for the ferromagnetic state, a solution for the antiferromagnetic state, and solutions for the spin glass state. The phase diagram, and the ground state energy and entropy are calculated. The ground state entropy of the model is shown to be a small positive quantity contrary to the Sherrington-Kirkpatrick infinitely long-ranged model. The phase diagram derived from the cluster approximation well explains the experimental phase diagrams, in particular, of fcc spin glass and the spin glass of EupSr1-pS. The distribution function for the spin glass in the pair approximation at T = 0 with a continuous distribution is obtained analytically. They are composed of δ-functions or of δ-functions and a quadratic continuous function.

The theory of the random Ising model formulated in terms of the distribution function of the effective field in the pair and in the cluster approximations is reviewed. An integral equation for the distribution function is derived. The integral equation has a solution for the paramagnetic state, a solution for the ferromagnetic state, a solution for the antiferromagnetic state, and solutions for the spin glass state. The phase diagram, and the ground state energy and entropy are calculated. The ground state entropy of the model is shown to be a small positive quantity contrary to the Sherrington­ Kirkpatrick infinitely long-ranged model. The phase diagram derived from the cluster approximation well explains the experimental phase diagrams, in particular, of fcc spin glass and the spin glass of EupSr1-pS. The distribution function for the spin glass in the pair approximation at T=O with a continu­ ous distribution is obtained analytically. They are composed of a-functions or of a-functions and a quadratic continuous function. In this paper the treatment of random spin systems, especially of the spin glass problem by the method of the distribution function of the effective field developed by our group is reviewed. A cluster (pair, triangle, square, tetrahedron, etc.) is taken in a given lattice (square, triangle, simple cubic, hexagonal, face-centered cubic, etc.). An effective field with a distribution is assumed to be applied to each vertex of the cluster. The partition function of the cluster is calculated exactly in terms of these effective fields. Self-consistent relation leads to an integral equation for the distribution function of the effective fields. The solution of the integral equation gives the physical quantities, the phase diagrams and the ground state properties. We consider random Ising models on a given crystal lattice. The hamiltonian &, the density matrix p, and the free energy Fare given by

This special issue of Journal of Physics A: Mathematical and Theoretical collects papers by speakers and participants of the conference `Viewing the World through Spin Glasses', held in Oxford (UK) on 31 August and 1 September 2007 in honour of Professor David Sherrington. It also includes contributions by many other active researchers in the field of spin glasses and related problems.The theory of spin glasses has a history of more than 30 years and continues to develop within itself as well as into an unexpectedly vast range of interdisciplinary subjects, including neural networks, error-correcting codes, optimization problems and social problems. Most of these amazing developments have their formal basis in the ground-breaking work of David Sherrington with Scott Kirkpatrick, centred on the SK model and the techniques devised to analyse it via the replica method. In this 'classic-of-classics' paper, a theoretical paradigm was suddenly established which became the common tool of analysis for thousands of papers in the following decades. It also led to deep developments in probability theory, through the efforts to understand the enigmatic Parisi solution of the SK model. The work of Professor Sherrington will continue to be an infinite source of our inspiration in many years to come.The purpose of the conference `Viewing the World through Spin Glasses' was to provide an overview of the present status of the fields which Professor Sherrington initiated, on the occasion of his 65th birthday, organized by John Cardy, Juan P Garrahan and the present Guest Editors. The first contribution in this special issue, by Professor Paul Goldbart, reflects his salute delivered at the conference dinner, and conveys its atmosphere very well. The papers that follow, ordered by the date of acceptance, represent the current activities of leading researchers in spin glasses and related fields, and we expect these to serve as milestones for future developments.We thank all the authors of this special issue and the participants of the conference for their valuable contributions.

The zero-point entropy of a spin glass is a difficult property to experimentally determine and interpret. Spin glass theory provides various predictions, including unphysical ones, for the value of the zero-point entropy, however experimental results have been lacking. We have investigated the magnetic properties and zero-point entropy of two spinel Cu2+ based spin glasses, CuGa2O4 and CuAl2O4. Dc- and ac-susceptibility and specific heat measurements show many characteristic spin glass features for both materials. The spin glass freezing temperature is determined to be Tf = 2.89 ± 0.05 K for CuGa2O4 and Tf = 2.30 ± 0.05 K for CuAl2O4. By integrating the specific heat data we have found that CuGa2O4 and CuAl2O4 have zero-point entropies of S0 = 4.96 JK-1mol-1 and S0 = 4.76 JK-1mol-1 respectively. These values are closest to the prediction for a Sherrington-Kirkpatrick XY spin glass, however they are notably higher than all of the theoretical predictions. This indicates that CuGa2O4 and CuAl2O4 have a greater degeneracy in their ground states than any of the spin glass models.

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

When we called for contributions to this special issue we stressed the fact that we were interested in having the topic interpreted broadly. We asked for contributions ranging from equilibrium and dynamical studies of spin glasses, glassy behaviour in amorphous materials and low temperature physics, to applications in non-conventional areas such as error-correcting codes, image analysis and reconstruction, optimization and algorithms based on statistical mechanical ideas. This was because we believe that we have arrived at a very exciting moment for the development of this multidisciplinary approach, and that this issue should bear witness to, and summarize, such an exciting situation. Even a cursory look at the index of this issue shows, we believe, that our hopes have been completely fulfilled; we have a large variety of papers giving new insights into the whole range of fields. Our hope is that it will play a double role. On the one hand it will carry, as good journals always should, a number of good physics papers containing important results. On the other it should be a good summary of the state of the art for some time. The larger section of the issue is about slow dynamics. This is understandable, since slow dynamics is such a ubiquitous phenomenon. Here recent progress deals with glasses, spin glasses and far more general situations. Modifications of the celebrated fluctuation--dissipation theorem also play a crucial role. Finite dimensional systems (mainly spin glasses) are attracting a lot of attention since their behaviour is not yet well defined from a theoretical point of view. Here we have papers about Ising, Heisenberg and Potts spin glasses, together with the discussion of different types of disorder. Even if the mean field theory of spin glasses is well understood, important questions (about, for example, the value of the complexity and the detailed nature of the solution of the model) are still open, and they are discussed in this issue. This period has also seen very rich developments in rigorous results concerning complex disordered systems. We present some new rigorous results in this issue. As we have said before, we have focused on the strong paradigmatic and interdisciplinary nature of the recent developments in the subject. Maybe the spin glass theory is more important to some people since it allows us to study error-correcting codes and similar problems than because of the study of the spin glass materials themselves. Here we present a number of new results in many of these emerging directions. We then have new developments in image processing. One uses the mean field theory, the Bethe approximation and ideas from the dynamical approach. Also very important is the relation among statistical mechanics of disordered systems and optimization. We have here new results about colouring and the analysis of disordered systems ground states, together with a short review on vertex covering. The same ideas applied to codes are also finding many applications; here we have new work about low-density parity check codes and CDMA multi-user detection codes. Finally we have new results about the application of statistical mechanical ideas to game theory and to the so-called econophysics. We believe that this topic is also experiencing a fast and solid progress, and we are happy to be able to witness it here. We thank the authors who have been collaborative, open-minded toward improvements and punctual (as much as one could hope). We also appreciate the efforts of the referees who have worked hard towards ensuring high quality; we believe they have succeeded. We thank all the staff of Journal of Physics A: Mathematical and General for their exceptional work. Without all that this issue would not have been possible.

The authors solve, subject to the validity of some reasonable assumptions, the 'fully frustrated' Ising model in the limit of infinite dimensions using an extension of the TAP theory for spin glasses. In contrast to the TAP theory of the infinite-range spin glass, an infinite summation of diagrams is required to recover the Gibbs free energy for this model. The model undergoes a first-order transition. The method used to solve the model should have many applications to other physical problems.

This chapter discusses short-range spin glasses. It considers realistic spin glass models, in particular the Edwards–Anderson (EA) model. Both the EA and the Sherrington–Kirkpatrick (SK) models are idealizations of the complicated spatial structure of spin–spin interactions in real materials. In the EA idealization, the interactions are extremely short range, occurring only between spins that are nearest neighbors in the atomic lattice. This caricatures the actual spatial structure of laboratory spin glasses, but the EA model is nevertheless believed to distill their essential physics. In contrast, the SK model is bereft of geometric structure and behaves like an EA model in the limit of infinite dimension. Since we are really interested in finite dimensions, it is important to understand not only how the phenomena exhibited by the EA model depend on dimension d, but also how the similarities and differences between the EA and SK models depend on dimension.

: This paper is the second in a series by the authors concerned with the theory of viscosity solutions Hamilton-Jacobi equations in infinite dimensional spaces. The first paper introduced a notion of viscosity solution appropriate for the study of Hamilton-Jacobi equations in spaces with the so-called Radon-Nikodym property and obtained uniqueness theorems under assumptions paralleling the finite dimensional theory. The main results of the current paper concern existence of solutions of stationary and time-dependent Hamilton-Jacobi equations. In order to establish these results it is necessary to overcome the difficulties associated with the fact that bounded sets are not precompact in infinite dimensions and this is done by sharp constructive estimates coupled with the use of differential games to solve regularized problems. Interest in this subject arises on the abstract side from the desire to contribute to the theory of linear partial differential equations in infinite dimensional spaces to treat natural questions raised by the finite dimensional theory. Interest also arises from potential applications to the theory of control of partial differential equations. However, the results herein do not apply directly to problems of the form arising in the control of partial differential equations, a question which wil be treated in the next paper of the series. Additional keywords: Banach spaces, Existence theory. (Author)

Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions

The effect of correlations in disorder variables is a largely unexplored topic in spin glass theory. We study this problem through a specific example of correlated disorder introduced in the Ising spin glass model. We prove that the distribution function of the magnetization along the Nishimori line in the present model is identical to the distribution function of the spin glass order parameter in the standard Edwards-Anderson model with symmetrically distributed independent disorder. This result means that if the Edwards-Anderson model exhibits replica symmetry breaking, the magnetization distribution in the correlated model has support on a finite interval, in sharp contrast to the conventional understanding that the magnetization distribution has, at most, two delta peaks. This unusual behavior challenges the traditional argument against replica symmetry breaking on the Nishimori line in the Edwards-Anderson model. In addition, we show that when temperature chaos is present in the Edwards-Anderson model, the ferromagnetic phase is strictly confined to the Nishimori line in the present model. These findings are valid not only for finite-dimensional systems but also for the infinite-range model, and highlight the need for a deeper understanding of disorder correlations in spin glass systems.

Large deviation for Markov processes can be studied by Hamilton--Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a method applied to a class of Hilbert-space-valued small diffusion processes. The examples include stochastically perturbed Allen--Cahn, Cahn--Hilliard PDEs and a one-dimensional quasilinear PDE with a viscosity term. We prove the comparison principle using a variant of the Tataru method. We also discuss different notions of viscosity solution in infinite dimensions in such context.

An effective-field theory of the spin glass is presented, using only the conventional molecular-field approximation, in order to make the so-called replica formalism of the spin glass transparent.

Determination of the precise location of the multicritical point and phase boundary is a target of active current research in the theory of spin glasses. In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.